Bayesian Estimation in Hierarchical Models

Bayesian data analysis involves describing data by meaningful mathematical models, and allocating credibility to parameter values that are consistent with the data and with prior knowledge. The Bayesian approach is ideally suited for constructing hierarchical models, which are useful for data structures with multiple levels, such as data from individuals who are members of groups which in turn are in higher-level organizations. Hierarchical models have parameters that meaningfully describe the data at their multiple levels and connect information within and across levels. Bayesian methods are very flexible and straightforward for estimating parameters of complex hierarchical models (and simpler models too). We provide an introduction to the ideas of hierarchical models and to the Bayesian estimation of their parameters, illustrated with two extended examples. One example considers baseball batting averages of individual players grouped by fielding position. A second example uses a hierarchical extension of a cognitive process model to examine individual differences in attention allocation of people who have eating disorders. We conclude by discussing Bayesian model comparison as a case of hierarchical modeling.